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The power rule is a basic technique used for differentiating functions that are raised to a power. In its simplest form, the power rule states that for any real number n, if y = x^n, then the derivative of y with respect to x is ny^{n-1}. Stated mathematically, this is:
\[ \frac{d}{dx}(x^n) = nx^{n-1} \].
This rule greatly simplifies the process of differentiation because it allows us to quickly find the derivative without resorting to the limit definition of a derivative each time. In the exercise provided, the power rule was applied in Step 2 after recognizing that the outer function can be expressed as u raised to the eighth power. By taking the derivative with respect to the inner function u, we get 8u^7, which follows directly from the power rule.

The quotient rule is essential for finding the derivative of a function that is the quotient of two differentiable functions. This rule states that if you have a function h(x) = f(x) / g(x), where both f(x) and g(x) are differentiable, the derivative h'(x) is given by:
\[ h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} \].
This formula may appear complex, but it essentially says that the derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the square of the denominator. In the initial problem, to differentiate the inner function u, which is a quotient of an exponential function and a linear function, the quotient rule comes into play, as shown in Step 3. It is critical to accurately differentiate the numerator and the denominator separately before applying the rule itself.

Differentiating exponential functions requires understanding the unique property of the exponential function. The derivative of an exponential function is proportional to the function itself. For any base a, the derivative of a^x is a^x times the natural logarithm of the base a. The most common exponential function in calculus is e^x, which has a derivative that is simply e^x itself. Formally, if y = e^x, then:
\[ \frac{d}{dx}(e^x) = e^x \].
In our exercise case, this simple derivative is part of the inner function u as indicated in Step 3. Whenever differentiating an exponential term like e^x, remember this unique property. This rule is also known as the Exponential Rule and plays a key role in simplifying many calculus problems dealing with exponential growth or decay.